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How to integrate ln(x)/x?

To integrate ln(x)/x, use substitution u = ln(x) and du/dx = 1/x.

Integrating ln(x)/x can be done using substitution. Let u = ln(x), then du/dx = 1/x. Rearranging, we have dx = x du. Substituting these into the integral, we get:

∫ ln(x)/x dx = ∫ u du

Integrating u with respect to u gives:

∫ ln(x)/x dx = u + C

Substituting back u = ln(x), we get:

∫ ln(x)/x dx = ln(x) + C

Therefore, the integral of ln(x)/x is ln(x) + C, where C is the constant of integration.

It is important to note that the domain of ln(x)/x is (0, ∞), as ln(x) is undefined for x ≤ 0. Also, ln(x) approaches negative infinity as x approaches 0, so the integral ln(x)/x diverges at x = 0.

For more detailed exploration on integration techniques used in this solution, refer to the page on Techniques of Integration.

A-Level Maths Tutor Summary: To find the integral of ln(x)/x, use substitution with u = ln(x), turning the integral into a simpler form, ∫ u du. This simplifies to ln(x) + C, where C is a constant. Remember, this only works for x greater than 0 because ln(x) isn't defined for x ≤ 0. So, the answer is ln(x) + C, but be mindful of where it applies. Explore more on the foundational concepts in Basic Integration Rules and delve into both Definite and Indefinite Integrals for a broader understanding.

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